TY - JOUR

T1 - Ramanujan's class invariants, Kronecker's limit formula, and modular equations

AU - Berndt, Bruce C.

AU - Chan, Heng Huat

AU - Zhang, Liang Cheng

PY - 1997

Y1 - 1997

N2 - In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, b'ut approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of , class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan's class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker's limit formula, the second employs modular equations, and the third uses class field theory to make Watson's "empirical method"rigorous.

AB - In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, b'ut approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of , class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan's class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker's limit formula, the second employs modular equations, and the third uses class field theory to make Watson's "empirical method"rigorous.

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U2 - 10.1090/s0002-9947-97-01738-8

DO - 10.1090/s0002-9947-97-01738-8

M3 - Article

AN - SCOPUS:21744438349

SN - 0002-9947

VL - 349

SP - 2125

EP - 2173

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 6

ER -